Group theory is a mathematical formalism which has important applications in chemistry, physics and geology. In particular, the symmetry operations, namely rotations, reflections, inversions, reflection-rotations, and identity, inherent in molecular structures, constitute mathematical groups. These mathematical groups are known as point groups because one or more points at the center of the structure are unshifted by all of the operations of the group. Each point group has an overall order, equal to the number of times the basic asymmetric unit, or any random point, of a symmetrical structure will be repeated. This overall order also equals the number of symmetry operations associated with a given point group. Group theory and classification schemes for assigning molecules to their point groups are taught in advanced undergraduate and/or beginning graduate courses in the chemistry and physics curricula of modern universities.
Many important properties of various organic and inorganic molecules can be ascertained through assigning point groups comprised of three dimensional symmetry operations to those molecules. For example, depending on the point group assigned to a given molecule based on the number of symmetry operations found for that molecule, one can determine the presence or absence of a dipole moment in the molecule. Whether the molecule is optically active can be determined using point group symmetry. Point group symmetry may also be used to evaluate whether certain molecules require resonance structures for an accurate valence bond representation of bonding and also whether the atomic orbitals in particular orientations of certain molecules will have a finite or zero net overlap. In short, point group symmetry has numerous beneficial applications in evaluate and classifying various organic and inorganic molecules, while requiring only analysis of certain symmetrical attributes of the structure of the molecule.
The teaching of the principles of point group symmetry is difficult because specific molecules belonging to various point groups have very disparate structures and their three dimensional structures are often difficult to represent in two dimensions. Thus, accurate representations in textbooks of various structures and molecules exhibiting certain point groups are not readily achievable.
Various molecular model sets constructed from numerous materials are well-known and commonly used teaching tools for illustrating molecular structures, mainly the relative positions of the constituent atoms in the molecule. Their utility is not readily extended to the systematic illustration of the symmetry point groups, because in most molecules, some atoms sit on the symmetry axes, planes or centers, and therefore are fewer in number than the full order of the group. For example, the molecule AuCI.sub.4.sup.- is in the point group D.sub.4 h, that has an overall order of 16, but the molecule itself has only four chlorides lying off the principal four-fold axis of the point group.